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Conic degeneration and the determinant of the Laplacian

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Abstract

We investigate the behavior of various spectral invariants, particularly the determinant of the Laplacian, on a family of smooth Riemannian manifolds Ω ge that undergo conic degeneration, i.e., that converge in a particular way to a manifold with a conical singularity. Our main result is an asymptotic formula for the determinant up to terms that vanish as ∈ goes to 0. The proof proceeds in two parts: we study the fine structure of the heat trace on the degenerating manifolds via a parametrix construction, and then use that fine structure to analyze the zeta function and determinant of the Laplacian.

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References

  1. P. Albin, A renormalized index theorem for some complete asymptotically regular metrics: the Gauss-Bonnet theorem, Adv. Math., 213 (2007), 1–52.

    Article  MATH  MathSciNet  Google Scholar 

  2. C. Anné and J. Takahashi, Partial collapsing and the spectrum of the Hodge Laplacian. arXiv:1007.2949.v5[math.DG].

  3. M. van den Berg and S. Srisatkunarajah, Heat equation for a region in ℝ2 with a polygonal boundary, J. London Math. Soc. (2) 37 (1988), 119–127.

    MATH  MathSciNet  Google Scholar 

  4. J. Bruning and R. Seeley, The expansion of the resolvent near a singular stratum of conic type, J. Funct. Anal. 95 (1991), 255–290.

    Article  MathSciNet  Google Scholar 

  5. J. Cheeger, Analytic torsion and the heat equation, Ann. of Math. (2) 109 (1979), 259–322.

    Article  MATH  MathSciNet  Google Scholar 

  6. J. Cheeger, Spectral geometry of singular Riemannian spaces, J. Differential Geom. 18 (1983), 575–657.

    MATH  MathSciNet  Google Scholar 

  7. Y. Ding, Heat kernels and Green’s functions on limit spaces, Comm. Anal. Geom. 10 (2002), 475–514.

    MATH  MathSciNet  Google Scholar 

  8. C. Epstein, R. B. Melrose, and G. Mendoza, Resolvent of the Laplacian on strictly pseudoconvex domains, Acta Math. 167 (1991), 1–106.

    Article  MATH  MathSciNet  Google Scholar 

  9. B. Fedosov, Asymptotic formulas for eigenvalues of the Laplacian in a polyhedron, Doklady Akad. Nauk SSSR, 157 (1964), 536–538.

    MathSciNet  Google Scholar 

  10. D. Grieser, Basics of the b-calculus, in Approaches to Singular Analysis, Birkhäuser, Basel, 2001, pp. 30–84.

    Chapter  Google Scholar 

  11. C. Guillarmou, Resolvent at low energy and Riesz transform for differential forms, unpublished.

  12. C. Guillarmou and A. Hassell, Resolvent at low energy and Riesz transform for Schrödinger operators on asymptotically conic manifolds. I, Math. Ann. 341(2008), 859–896.

    Article  MATH  MathSciNet  Google Scholar 

  13. C. Guillarmou and A. Hassell, Resolvent at low energy and Riesz transform for Schrödinger operators on asymptotically conic manifolds. II. Ann. Inst. Fourier (Grenoble) 59 (2009), 1553–1610.

    Article  MATH  MathSciNet  Google Scholar 

  14. C. Guillarmou and D. Sher, Low energy resolvent for the Hodge laplacian: applications to Riesz transform, Sobolev estimates, and analytic torsion, Int. Math. Res. Not. (2014) (rnu119).

  15. H. Khuri, Heights on the moduli space of Riemann surfaces with circle boundaries, Duke Math. J. 64 (1991), 555–570.

    Article  MATH  MathSciNet  Google Scholar 

  16. Young-Heon Kim, Surfaces with boundary: their uniformizations, determinants of Laplacians, and isospectrality, Duke Math. J. 144 (2008), 73–107.

    Article  MATH  MathSciNet  Google Scholar 

  17. R. Mazzeo, Elliptic theory of differential edge operators, I, Comm. Partial Differential Equations 16 (1991), 1615–1664.

    Article  MATH  MathSciNet  Google Scholar 

  18. R. Mazzeo, Resolution blowups, spectral convergence and quasi-asymptotically conic spaces, Journées Équations aux Dérivées Partielles, Evian, 2006.

  19. R. Mazzeo and J. Rowlett, A heat trace anomaly on polygons, Math. Proc. Cambridge Philos. Soc., to appear.

  20. P. T. McDonald, The Laplacian for spaces with cone-like singularities. Ph.D. thesis, Massachusetts Institute of Technology, 1990.

  21. R B. Melrose, The Atiyah-Patodi-Singer Index Theorem, A. K. Peters, Ltd., Boston, MA, 1993.

    MATH  Google Scholar 

  22. R. B. Melrose, Calculus of conormal distributions on manifolds with corners, Internat. Math. Res. Notices 1992 (1992), 51–61.

    Article  MATH  MathSciNet  Google Scholar 

  23. R. B. Melrose, Differential analysis on manifolds with corners, in preparation; available online at http://math.mit.edu/~rbm/book.html.

  24. R. B. Melrose, and M. Singer, Scattering configuration spaces, arXiv:0808.2022.v1 math.DG].

  25. E. Mooers, The heat kernel for manifolds with conic singularities, Ph.D. thesis, Massachusetts Institute of Technology, 1996.

  26. W. Müller, Analytic torsion and R-torsion of Riemannian manifolds, Adv. Math. 28 (1978), 233–305.

    Article  MATH  Google Scholar 

  27. B. Osgood, R. Phillips, and P. Sarnak, Extremals of determinants of Laplacians, J. Funct. Anal. 80 (1988), 148–211.

    Article  MATH  MathSciNet  Google Scholar 

  28. B. Osgood, R. Phillips, and P. Sarnak, Compact isospectral sets of surfaces, J. Funct. Anal. 80 (1988), 212–234.

    Article  MATH  MathSciNet  Google Scholar 

  29. B. Osgood, R. Phillips, and P. Sarnak, Moduli space, heights and isospectral sets of plane domains, Ann. of Math. (2), 129 (1989), 293–362.

    Article  MATH  MathSciNet  Google Scholar 

  30. S. Rosenberg, The Laplacian on a Riemannian Manifold, Cambridge University Press, Cambridge, 1997.

    Book  MATH  Google Scholar 

  31. J. Rowlett, Spectral geometry and asymptotically conic convergence, Comm. Anal. Geom. 16 (2008), 735–798.

    Article  MATH  MathSciNet  Google Scholar 

  32. J. Rowlett, Thesis correction, unpublished; available online at www.analysis.uni-hannover.de/~rowlett/summary.html.

  33. D. B. Ray and I. M. Singer, R-torsion and the Laplacian on Riemannian manifolds, Adv. Math. 7 (1971), 145–210.

    Article  MATH  MathSciNet  Google Scholar 

  34. D. A. Sher, Conic degeneration and the determinant of the Laplacian, Ph. D. thesis, Stanford University, 2012.

  35. D, A. Sher, The heat kernel on an asymptotically conic manifold, Anal. PDE 6 (2013), 1755–1791.

    Article  MATH  MathSciNet  Google Scholar 

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  1. Department of Mathematics, University of Michigan, Ann Arbor, MI, 48109-1043, USA

    David A. Sher

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  1. David A. Sher
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Correspondence to David A. Sher.

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Sher, D.A. Conic degeneration and the determinant of the Laplacian. JAMA 126, 175–226 (2015). https://doi.org/10.1007/s11854-015-0015-3

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  • DOI: https://doi.org/10.1007/s11854-015-0015-3

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